\(QS25_{2}^{(0)}\)

Description

Topological configuration of singularities: \(s,a;S,N,N\)

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(25\)	\(41\)	\(221101\)

Example

The quadratic differential system

\[\begin{cases} \dot{x} = y+x^{2}+x \, y \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+y^{2}/4 \end{cases}\]

with parameters: \(e = 0.4\)

has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.

The phase portrait appears in the following papers

With name \(AA^s_5\) in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes, J. Dynam. Differential Equations { bf 33} (2021), no.~4, 1779--1821; MR4333383
With name \(11S4\) in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}
With name \(1S02\) in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).
With name \(Fig 5.129 S^2_{9,2}\) in {J. C. Artés, J. Llibre and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018, vi+267 pp.Note (for name \(Fig 5.129 S^2_{9,2}\)): The system has 1 limit cycle.
With name \(V11\) in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Topological classification of quadratic polynomial differential systems with a finite semi-elemental triple saddle, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 26} (2016), no.~11, 1650188, 26 pp.; MR3566296
With name \(S^2_{9,2}\) in {J. C. Artés, R. E. Kooij and J. Llibre}, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. { bf 134} (1998), no.~639, viii+108 pp.; MR1432139
With name \(V11\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.
With names \(Fig 1.3 f\), \(Fig 1.3 h\) and \(Fig 1.7 c\) in {J. W. Reyn and R. E. Kooij}, Phase portraits of non-degenerate quadratic systems with finite multiplicity two, Differential Equations Dynam. Systems { bf 5} (1997), no.~3-4, 355--414; MR1660222Note (for name \(Fig 1.3 h\)): The system has 1 limit cycle.

Neighbours of Codimension 1

Through the border \(QS27_{3}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
Through the border \(QS27_{4}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
Through the border \(QS45_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS13_{1}^{(0)}\).
Through the border \(QS25_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{1}^{(0)}\).
Through the border \(QS25_{5}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{3}^{(0)}\).
Through the border \(QS25_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{3}^{(0)}\).
Through the border \(QS38_{8}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{4}^{(0)}\).
Through the border \(QS38_{13}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{6}^{(0)}\).
Through the border \(QS38_{17}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{7}^{(0)}\).
Through the border \(QS38_{19}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{9}^{(0)}\).
Through the border \(QS38_{23}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{11}^{(0)}\).
Through the border \(QS38_{26}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{13}^{(0)}\).
Through the border \(QS38_{27}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{13}^{(0)}\).

Comments

This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS25_{2}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.